Corrections to the First Printing

Transcription

1 Corrections to the First Printing Chapter 2 (i) Page 48, Paragraph 1: cells/µ l should be cells/µl without the space. (ii) Page 48, Paragraph 2: Uninfected cells T i should not have the asterisk. Chapter 3 (i) Page 52: Equation (3.4) should read z(t, ω) = e a(ω)t [z 0 (ω) + (ii) Page 55: Last Paragraph: Remove comma after copper. t 0 ] e a(ω)s b(s, ω)ds (iii) Page 57: The reference [50] should be cited for the Neutron Diffusion example. (iv) Page 61: In (3.39), the index should read j = 1,..., n rather than t j = 1,..., n. Two sentences later, it should read Note that ν and N respectively designate the dimension of the response and the number of states. Chapter 4 (i) Page 73: Karl Person should be Karl Pearson. (ii) Page 79: To clarify, consider n random samples X 1,..., X n from a population with pdf g(x). They are identically distributed if g(x) = f X1 (x 1 ) =... = f Xn (x n ). For consistency, (4.20) should then read n f X (x 1,, x n ) = f Xi (x i ). (iii) Page 81, Line 6: A (1 α) 100% confidence interval should read The 95.45% confidence interval. (iv) Page 81, Example Note that 1 α/2 is the probability rather than the interval. (v) Pages 83: The likelihood function at the bottom of the page should be L(q υ) = i=1 n f Υi (υ i ; q). This same expression should be used in the definition of the MLE on page 84. (vi) Page 84: The condition before (4.29) should read q (l(q, σ2 Q) = 0. (vii) Page 90: To compare to the previous definition of a random variable, we assume here that there is an ordering on S. This does not have be true in general, in which case one would modify the measure-theoretic definition. For this reason, a general S-valued random variable does not necessarily have a mean. i=1 1

2 (viii) Page 92: The percentage of teams who win/lose their next game is [0.64, 0.36]. (ix) Page 97: The solution is z(t, ω). (x) Page 99: The number 100 is never used and can be replaced by a large number of. (xi) Page 100: We added the following sentence to the footnote. For example, θ is typically used to denote calibration parameters in statistics whereas q is commonly employed in the mathematics literature. (xii) Page 102: Replace mean by mode since the latter is easier to distinguish. (xiii) Page 103: The observations in Example 4.69 should be υ = [υ 1,..., υ n ] rather than υ = [υ 1,..., υ N ]. (xiv) Page 105: In Exercise 4.6, the variances should be σ 2 z = 1/3 and σ 2 x = (b a) 2 /12. Chapter 5 (i) Page 111: In Example 5.2, we should have φ 1 (x) = 1, λ 1 = length(d). (ii) Pages : To remain consistent with equation (5.1), the variables u, u l, u r and u 0 should be T, T l, T r and T 0. Chapter 6 (i) Pages : The discussion is clarified if one refers to the parameter set when defining the concepts of identifiability and influential parameter spaces. (ii) Page 114, Figure 6.1(b): For consistency, the parameters values in the figure and first paragraph should be designated q 1 and q 2 rather than q 1 and q 2, which can be confused with the parameter components. (iii) Page 115, Paragraph 2: This should be K = k m. (iv) Page 117: In the line following (6.3), this should read Σ R n p. (v) Page 119: Algorithm 6.10 (Random Range Finder). (vi) Page 120: The first line should have t i = (i 1) t, t = 1 n 1, i = 1,..., n. (vii) Page 120: In Case i, the pivoted QR factorization should be A T P = QR. (viii) Pages : The rank of A and dimension of the identifiable subspace should be 49 rather than 50. Chapter 7 (i) Page 132: We added the following sentence to the end of the first paragraph. Finally, we note that parameters are often denoted by θ in the statistics literature. (ii) Page 141: Equation (7.9) should be cited after (7.29) rather than (7.3). 2

3 (iii) Pages : Subscript c should be subscript C. (iv) Page 153, Exercise 7.8: This should be two standard deviations. Chapter 8 (i) Page 155: we added the following paragraph in the introduction to the chapter. We remind readers that calibration parameters and observed data are commonly denoted by θ and y in the statistics literature. It is also common to employ the same notation for the random variable and realization and let the context dictate the meaning. (ii) Page 158: Equation (8.6) should read π(q υ) 1 k i=1 e (SS ζ i SS q)/2σ 2 0w i. (iii) Page 164: The Jeffreys prior should read π 0 (q, σ 2 ) = 1 σ 2. (iv) Page 164: In Algorithm 8.5, (a) and (g) should read (a) Sample z k N(0, I p ) and (g) Update s 2 k Inv-gamma(a val, b val ). (v) Pages 165: The covariance estimate is V = s 2 0 [X T (q 0 s. s)x T (q 0 s. s)]. (vi) Page 167: Figure 8.6 has been updated so that the C chain and density match. (vii) Page 173: The first sentence should read In theory, cov(q 0,, q k 1 ) can be computed using the empirical covariance formula ( cov(q 0,, q k 1 ) = 1 k 1 ) q i (q i ) T k q k ( q k ) T k 1 where q k = 1 k 1 k i=0 qi and q i are column vectors. Similarly, in the equation following (8.20), q k should be q i in the first two lines. (viii) Page 175: In Algorithm 8.8, (a) and (g) should read (a) Sample z k N(0, I p ) and (g) Update s 2 k Inv-gamma(a val, b val ). (ix) Page 176: Step 2 should read Sample z k N(0, I p ). (x) Page 176: The covariance estimate is V = s 2 0 [X T (q 0 s. s)x T (q 0 s. s)]. (xi) Page 182: In the caption of Figure 8.15, be should be b E. (xii) Pages : The parameter dimension should be changed from d to p to remain consistent with previous notation. i=0 3

4 Chapter 9 (i) Pages : The t-distributions should be modified to be t n p,1 α/2 to be consistent with previous notation. (ii) Page 200: The t-distributions are t n 2,1 α/2. (iii) Page 201: The caption of Figure 9.5 should be modified to reference the confidence and prediction intervals specified using the linear theory in Section Chapter 10 (i) Page 211: The basis function in (10.13) should be ψ i (q) rather than ψ(q). (ii) Page 211: In Example 10.3, we should have Q U( 1, 1). (iii) Page 215: Equation (10.22) should have the independent variable t rather than x, du = f(t, Q, u), t > 0 dt u(0, Q) = u 0. (iv) Page 216: The period is missing at the end of the first sentence in the subsection Stochastic Galerkin Method. (v) Page 217: Two lines before (10.33) should have q r as collocation points q m. (vi) Page 219: In the Stochastic Weak Formulation, the index range should be k = 0,..., K for Ψ k (Q) rather than k = 1,..., K. (vii) Page 226: In the first sentence of Section 10.3, the word simply should be simple. (viii) Page 227: Equation (10.52) should have the factor 1 γ i the equation following (10.54) should be n = 0. rather than γ i. The lower index in (ix) Pages : The heat equation (10.60) should have ordinary derivatives d2 u dx 2 partial derivatives. rather than (x) Page 235: The second expression in (10.64) should be var [ y K (ω F, Q) ] = K yk 2 (ω F )γ k. (xi) Page 235: Because we are considering Q 1, Q 2 and Q 3 as normally distributed random variables with mean 0 and standard deviation 1, N(0, 1), it is clearer if we write the density as ρ Q (q) = and the quadrature points as q r = [q r 1 1, q r 2 2, q r 3 3 ]. k=1 ( 1 2π ) 3 e q2 1 /2 e q2 2 /2 e q2 3 /2 (xii) Page 236: The x-label in Figure 10.4(a) should be ω F rather than ω. 4

5 Chapter 11 (i) Page 241: To be consistent with Table 11.1, the first equation should read I (1) f = f(q)dq 1 2 R f(q r )w r. (ii) Page 242: The upper limit and stepsize in the trapezoid rule should be corrected in (11.3) to yield [ ] Q (1) l f = h R l 2 l f(0) + f(1) + 2 f(ql r 2 ) The line following (11.4) should be changed to state The weights are thus [ h l 2, h l,, h l, h l 2 ]. (iii) Page 251: In the final two equations, the collocation points should be denoted by q j rather than q j. r=1 r=1 Chapter 12 (i) Page 263: In the line following (12.2), T source should be interpreted as a nonphysical source term rather than the source temperature. This is illustrated by the correct parameter estimates T source = o C, h = , η = at the bottom of the page. The footnote should be deleted. (ii) Page 264: In the second line, the parameters should be q = [T source, h, η]. We note that constraints must be placed on these parameters to address parameter identifiability for a single data set. (iii) Page 265: As illustrated in the posted code, the optimal parameters were obtained using the MATLAB routine fminsearch.m. Due to confounding between the physical and algebraic model components, different values will be obtained if one uses lsqnonlin.m. This indicates a number of close local minima. (iv) Page 269: In Exercise 12.1, it is easier to use the model (12.1) than (12.3). This is also true in Exercise 12.3 on Page 270. Chapter 13 (i) Page 275: The final limits in (13.9) should be j > i rather than j i. (ii) Page 276: We replaced q 1 and q 2 with q i and q j to provide general relations. (ii) Page 279: The lower limit in (13.23) should be k = 0. The expression in the third line after (13.23) should read Ψ k (q m ) = δ km. Chapter 15 (i) Page 328: In Example 15.8, Q 1 N(0, σ 2 1 ). 5

6 (ii) Page 334: In Example 15.14, k = 2 should be replaced with p = 2. Bibliography (i) The title of [3] should be Estimation and prediction with HIV-treatment interruption data. (ii) Citation [66] can be appended to include SIAM Journal on Scientific Computating, in press. 6